Evolution

Download e-book for iPad: Abstract Evolution Equations, Periodic Problems and by D Daners

By D Daners

ISBN-10: 0582096359

ISBN-13: 9780582096356

A part of the "Pitman study Notes in arithmetic" sequence, this article covers: linear evolution equations of parabolic style; semilinear evolution equations of parabolic kind; evolution equations and positivity; semilinear periodic evolution equations; and functions.

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Extra info for Abstract Evolution Equations, Periodic Problems and Applications

Sample text

It is clear by (U 4) that the uniqueness of an evolution operator follows from this formula. The above representation formula – which will be instrumental in the treatment of semilinear equations in Chapter IV – is usually referred to as the variation-of-constants formula. We will adhere to this terminology and use it without further reference. 6 Theorem There exists a unique evolution operator for the family A(t) U (t, s) L(Xi ) (i = 0, 1) and 0≤t≤T . Moreover, (t − s) A(t)U (t, s) are bounded uniformly in (t, s) ∈ ∆T with a constant only depending on M , ρ, the H¨ older −1 norm of A(·) and a bound for A(t)A (s) .

The norms on E0 and E1 will be denoted by · 0 and · 1 , respectively. 6) D(x; E) := {(x0 , x1 ) ∈ E0 × E1 ; x0 + x1 = x}. We now define a function K(· , · ; E): (0, ∞) × E0 → R+ by setting K(t, x; E) := inf{ x0 0 + t x1 1 ; (x0 , x1 ) ∈ D(x; E)}. This function is called the K-functional. Furthermore, for each x ∈ E0 , the function K(· , x; E) is increasing and concave. It is not difficult to prove that K(t, · ; E) t>0 is a family of norms on E0 which are all equivalent to · 0 . Let now θ ∈ (0, 1) and 1 ≤ p < ∞ and define for each x ∈ E0 the expressions x θ,p ∞ := t −θ K(t, x; E) p 0 1 p dt t , and x θ,∞ := sup t−θ K(t, x; E).

3 we exclude the case p = ∞ because E1 needs not to be dense in (E0 , E1 )θ,∞ . By the same results in [22] we invoked in the case p = ∞, all other properties of an admissible family are also valid if we let p = ∞ . We shall return to this when we treat the continuous interpolation functor. (b) We actually have imbeddings for different p’s (cf. 1(b)). (E0 , E1 )θ,p ⊂→ (E0 , E1 )θ,q for any θ ∈ (0, 1) and 1 ≤ p ≤ q ≤ ∞. Moreover, these imbeddings are dense if q < ∞ (cf. 2(b)). We also have the following imbeddings (cf.

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Abstract Evolution Equations, Periodic Problems and Applications by D Daners


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